• Corpus ID: 239024651

Large Salem Sets Avoiding Nonlinear Configurations

  title={Large Salem Sets Avoiding Nonlinear Configurations},
  author={Jacob Denson},
We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions tfi : pT qn ́2 Ñ Tu, we obtain a Salem subset of T with dimension d{pn ́ 1q avoiding nontrivial solutions to the equation xn ́ xn ́1 “ fipx1, . . . , xn ́2q. For a countable family of smooth functions tfi : pT qn ́1 Ñ Tu satisfying a modest geometric condition, we obtain a Salem… 

Figures from this paper


Large Sets Avoiding Patterns
We construct subsets of Euclidean space of large Hausdorff dimension and full Minkowski dimension that do not contain nontrivial patterns described by the zero sets of functions. The results are of
Cartesian products avoiding patterns
The pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns, such as
Large Sets Avoiding Rough Patterns
The pattern avoidance problem seeks to construct a set $X\subset \mathbb{R}^d$ with large dimension that avoids a prescribed pattern. Examples of such patterns include three-term arithmetic
Salem sets with no arithmetic progressions
We construct Salem sets in $\mathbb{R}/\mathbb{Z}$ of any dimension (including $1$) which do not contain any arithmetic progressions of length $3$. Moreover, the sets can be taken to be Ahlfors
Patterns in random fractals
We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study
Fourier dimension and avoidance of linear patterns
The results in this paper are of two types. On one hand, we construct sets of large Fourier dimension that avoid nontrivial solutions of certain classes of linear equations. In particular, given any
Fourier Analysis and Hausdorff Dimension
Preface Acknowledgements 1. Introduction 2. Measure theoretic preliminaries 3. Fourier transforms 4. Hausdorff dimension of projections and distance sets 5. Exceptional projections and Sobolev
On the Fourier dimension and a modification
We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier
Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular
Fourier Transforms of Measures and Algebraic Relations on Their Supports
Si la transformee de Fourier d'une mesure decroit rapidement alors le support ne satisfait que tres peu des relations algebriques.